Legendre addition theorem

The spectroscopic measurements determine values for , [Pg.86]

For this situation of a transversely isotropic aggregate of transversely isotropic units, the Legendre addition theorem gives... 

The ° mn coefficients are the mean values of the generalized spherical harmonics calculated over the distribution of orientation and are called order parameters. These are the quantities that are measurable experimentally and their determination allows the evaluation of the degree of molecular orientation. Since the different characterization techniques are sensitive to specific energy transitions and/or involve different physical processes, each technique allows the determination of certain D mn parameters as described in the following sections. These techniques often provide information about the orientation of a certain physical quantity (a vector or a tensor) linked to the molecules and not directly to that of the structural unit itself. To convert the distribution of orientation of the measured physical quantity into that of the structural unit, the Legendre addition theorem should be used [1,2]. An example of its application is given for IR spectroscopy in Section 4. 

Ap and As are the absorbances measured with p- and s-polarization, respectively, and A0 — (Ap + 2As)/3 is the structural absorbance spectrum that would be measured for an isotropic sample. The order parameter of the main chain can be determined using the Legendre addition theorem (Equation (24)). 

In the case of uniaxial cylindrical symmetry around the MD, the (Pt)m values calculated from different crystal planes can be related to the (JV) coefficients of any of the three crystallographic angles using the Legendre addition theorem (see Section 4). For example, the (Pt)c coefficients corresponding to the crystalline c-axis is given by [81]... 

Other advantages of working in terms of spherical harmonic functions are that for cases with fibre symmetry, the Legendre addition theorem can be used, and affords considerable algebraic simplifications (see for example Ref. 25), and that for lower symmetries, the treatment can readily be generalised. It should be mentioned that the exact definitions of P2 cos9), etc., and p 9), can differ in different treatments due to the adoption of different normalisation procedures (see, for example. Chapter 5, Section 5.2).)... 

With these assumptions cos 0r and cos 0r could be deduced directly from eqns. (13), where 0r refers to the orientation of the principal axis of the tensor corresponding to as. By assuming that this axis makes the same angle y(= 19° 12 ) with the chain axis as it would in the crystal phase it was possible to use the Legendre addition theorem to deduce Af 200 and M400 and hence cos d and cos 0. More formally, the whole calculation can be expressed in terms of the following particular example of eqn. (14) ... 

Call, J.I., Taylor, D.J.R. and Stepto, R.F.T. (2000) Computer simulation studies of molecular orientation in polyethylene networks orientation functions and the legendre addition theorem. Macromolecules, 33,4966. 

In fluorescence spectroscopy, the orientation distribution of the guest probe is not necessarily identical to the actual orientation of the polymer chains, even if it is added at very small concentrations (i.e., a probe with high fluorescence efficiency). As a matter of fact, it is generally assumed that long linear probes are parallel to the polymer main chain, but this is not necessarily the case. Nevertheless, if the relation between the distribution of the probe axes and those of the polymer axes is known, the ODF of the structural units can be calculated from that of the probe thanks to the Legendre s addition theorem. Finally, the added probe seems to be mainly located in the amorphous domains of the polymer [69] so that fluorescence spectroscopy provides information relative to the noncrystalline regions of the polymeric samples. 

To evaluate the averages like those in Eq. (4.78), it is very convenient to pass from cosines ((en)k) to the set of corresponding Legendre polynomials for which a spherical harmonics expansion (addition theorem)... 

The functions /0 and /j are originally defined in terms of the polar angle that is 0 = arccos(e h). Thus, before performing integration, one needs to transform both integrands to the same coordinate set. Doing this with the aid of the addition theorem for Legendre polynomials, one finds... 

Applying the addition theorem, the Legendre polynomials P/(cos0p) can be expressed in terms of products of the spherical harmonics as below... 

We now expand the Legendre polynomial Pt (cos Oip) using the spherical harmonic addition theorem,... 

Let d and denote the arguments of the direction O, and let s cos 0. Clearly, no is completely defined by the quantities n M, and The Legendre polynomial P,(mo) may be written in terms of these variables by the application of the addition theorem which involves the associated functions PJ. 

By the use of the addition theorem of Legendre polynomials it can be shown that... 

The conversion of the two-electron repulsion potential e /rij to a form involving each electron separately is accomplished by expansion in a series of Legendre polynomials (cos co) where co is the angle subtended by the two electrons. Each P/, can be expressed in turn in a series of products of spherical harmonics by the spherical harmonic addition theorem ... 

The collision is also specified by the initial momentum that we write as the wave vector k. We can refer the direction of the vector R to the initial direction of the vector k. The addition theorem of spherical harmonics (Zare, 1988), where the Pi cosO)s are the Legendre polynomials,... 

In Eq. (7.2) the angle 6 is the velocity of the product with respect to the electrical field of the dissociation laser. By using the addition theorem for Legendre polynomials, Eq. (7.2), one can write for the pump probe Doppler profile of the angular distribution... 

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